The most promising means of resolving any scientific problem is the historical approach to it. In our case this approach proves a very essential one. The fact is that what are now called logic are doctrines that differ considerably in their understanding of the boundaries of this science. Each of them, of course, lays claim not so much simply to the title as to the right to be considered the sole modern stage in the development of world logical thought. That, therefore, is why we must go into the history of the matter.

The term ‘logic’ was first introduced for the science of thinking by the Stoics, who distinguished by it only that part of Aristotle’s actual teaching that corresponded to their own views on the nature of thinking. The term itself was derived by them from the Greek word logos (which literally means ‘the word’), and the science so named was very closely related to the subject matter of grammar and rhetoric. The mediaeval scholastics, who finally shaped and canonised the tradition, simply converted logic into a mere instrument (organon) for conducting verbal disputes, a tool for interpreting the texts of the Holy Writ, and a purely formal apparatus. As a result not only did the official interpretation of logic become discredited, but also its very name. The emasculated ‘Aristotelean logic’ therefore also became discredited in the eyes of all leading scientists and philosophers of the new times, which is the reason why most of the philosophers of the sixteenth to eighteenth centuries generally avoided using the term ‘logic’ as the name for the science of thought intellect, and reason.

Recognition of the uselessness of the official, formal, scholastic version of logic as the organon of real thought and of the development of scientific knowledge was the leitmotif of all the advanced, progressive philosophers of the time. ‘The logic now in use serves rather to fix and give stability to the errors which have their foundation in commonly received notions than to help the search after truth. So it does more harm than good,’ Francis Bacon said [Novum Organum] ‘I observed in respect to Logic,’ said Descartes, ‘that the syllogisms and the greater part of the other teaching served better in explaining to others those things that one knows (or like the art of Lully, in enabling one to speak without judgment of those things of which one is ignorant) than in learning what is new.’ [Discourse on Method] John Locke suggested that ‘syllogism, at best, is but the Art of fencing with the little knowledge we have, without making any Addition to it.’. [An Essay Concerning Human Understanding] On this basis Descartes and Locke considered it necessary to classify all the problems of the old logic in the sphere of rhetoric. And insofar as logic was preserved as a special science, it was unanimously treated not as the science of thinking but as the science of the correct use of words, names, and signs. Hobbes, for example, developed a conception of logic as the calculation of word signs.

In concluding his Essay Concerning Human Understanding, Locke defined the subject matter and task of logic as follows: ‘The business [of logic] is to consider the nature of signs the mind makes use of for the understanding of things, or conveying its knowledge to others.’ He treated logic as ‘the doctrine of signs’, i.e. as semiotics.

But philosophy, fortunately, did not jell at that level. The best brains of the period understood very well that it might be all right for logic to be interpreted in that spirit, but not for the science of thinking. True, in general, the representatives of purely mechanistic views of the world and of thinking held such a view of logic. Since they interpreted objective reality in an abstract, geometrical way (i.e. only purely quantitative characteristics were considered objective and scientific), the principles of thinking in mathematical science merged in their eyes with the logical principles of thinking in general, a tendency that took final form in Hobbes.

The approach of Descartes and Leibniz was much more careful. They too took to the idea of creating a ‘universal mathematics’ in place of the old, ridiculed, and discredited logic; and they dreamed of instituting a universal language, a system of terms strictly and unambiguously defined, and therefore admitting of purely formal operations in it.

Both Descartes and Leibniz, unlike Hobbes, were well aware of the difficulties of principle standing in the way of realising such an idea. Descartes understood that the definition of terms in the universal language could not be arrived at by amicable agreement, but must only be the result of careful analysis of the simple ideas, the bricks, from which the whole intellectual edifice of man was built; and that the exact language of ‘universal mathematics’ could only be something derived from ‘true philosophy’. Only then would one succeed in replacing thinking about the things given in reflection or imagination (i.e. in the terminology of the day, in contemplation) and in general in people’s real sense experience by a kind of calculus of terms and statements, and in drawing conclusions and inferences as infallible as the solutions of equations.

In supporting this point of Descartes’, Leibniz categorically limited the field of application of the ‘universal mathematics’ solely to those things that belonged to the sphere of the powers of imagination. The ‘universal mathematics’ should also, in his view, be only (so to say) a logic of the powers of imagination. But that was precisely why all metaphysics was excluded from its province, and also such things as thought, and action, and the field of ordinary mathematics, commensurate only in reason. A very essential reservation! Thought, in any case, thus remained outside the competence of the ‘universal mathematics’.

It is not surprising that Leibniz, with unconcealed irony, classified Locke’s treatment of logic, by which it was understood as a special doctrine of signs, as purely nominalist. Leibniz revealed the difficulties associated with such an understanding of logic. Above all, he said, the ‘science of reasoning, of judgments and inventions, seems very different from recognition of the etymologies and usage of words, which is something indeterminable and arbitrary. One must, moreover, when one wants to explain words, make an excursion into the sciences themselves as was seen in dictionaries; and one must not, on the other hand, engage in a science without at the same time giving a definition of the terms.’

Instead of the threefold division of philosophy into different sciences (logic, physics, and ethics) that Locke had taken over from the Stoics, Leibniz therefore suggested speaking of three different aspects, under which the same knowledge, the same truth, would function, namely theoretical (physics), practical (ethics), and terminological (logic). The old logic thus corresponded simply to the terminological aspect of knowledge, or, as Leibniz put it, ‘arrangement by terms, as in a handbook’. Such a systematisation, of course, even the best, was not a science of thought, because Leibniz had a more profound appreciation of thinking. And he classed the true doctrine of thought as metaphysics, in this sense following Aristotle’s terminology and the essence of his logic, and not the Stoics.

But why should thought be investigated within the framework of ‘metaphysics’? It was not a matter, of course, of indicating to which ‘department’ the theoretical understanding of thought ‘belonged’, but of a definite way of approaching the solution of an essential philosophical problem. And the difficulty constantly facing every theoretician lies in understanding what it is that links knowledge (the totality of concepts, theoretical constructions, and ideas) and its subject matter together, and whether the one agrees with the other, and whether the concepts on which a person relies correspond to something real, lying outside his consciousness? And can that, in general, be tested? And if so, how?

The problems are really very complicated. An affirmative answer, for all its seeming obviousness, is not quite so simple to prove, and as for a negative answer, it proves possible to back it up with very weighty arguments, such as that, since an object is refracted in the course of its apprehension through the prism of the ‘specific nature’ of the organs of perception and reason, we know any object only in the form it acquires as a result of this refraction. The ‘existence’ of things outside consciousness is thus by no means necessarily rejected. One thing ‘only’ is rejected, the possibility of verifying whether or not such things are ‘in reality’ as we know and understand them. It is impossible to compare the thing as it is given in consciousness with the thing outside consciousness, because it is impossible to compare what I know with what I don’t know, what I do not see, what I do not perceive, what I am not aware of. Before I can compare my idea of a thing with the thing, I must also be aware of the thing, i.e. must also transform it into an idea. As a result I am always comparing and contrasting only ideas with ideas, although I may think that I am comparing the idea with the thing.

Only similar objects, naturally, can be compared and contrasted. It is senseless to compare bushels and rods, poles, or perches, or the taste of steak and the diagonal of a square. And if, all the same, we want to compare steaks and squares, then we will no longer be comparing ‘steak’ and ‘square’ but two objects both possessing a geometrical, spatial form. The ‘specific’ property of the one and of the other cannot in general be involved in the comparison.

‘What is the distance between the syllable A and a table? The question would be nonsensical. In speaking of the distance of two things, we speak of their difference in space.... Thus we equalise them as being both existences of space, and only after having them equalised sub specie spatii [under the aspect of space] we distinguish them as different points of space. To belong to space is their unity.’ In other words, when we wish to establish a relation of some sort between two objects, we always compare not the ‘specific’ qualities that make one object ‘syllable A’ and the other a ‘table’, ‘steak’, or a ‘square’, but only those properties that express a ‘third’ something, different from their existence as the things enumerated. The things compared are regarded as different modifications of this ‘third’ property common to them all, inherent in them as it were. So if there is no ‘third’ in the nature of the two things common to them both, the very differences between them become quite senseless.

In what are such objects as ‘concept’ (‘idea’) and ‘thing’ related? In what special ‘space’ can they be contrasted, compared, and differentiated? Is there, in general, a ‘third’ thing in which they are ‘one and the same’, in spite of all their directly visible differences? If there is no such common substance, expressed by different means in an idea and in a thing, it is impossible to establish any intrinsically necessary relationship between them. At best we can ‘see’ only an external relation in the nature of that which was once established between the position of luminaries in the heavens and events in personal lives, i.e. relations between two orders of quite heterogeneous events, each of which proceeds according to its own, particular, specific laws. And then Wittgenstein would be right in proclaiming logical forms to be mystical and inexpressible.

But in the case of the relationship between an idea and reality there is yet another difficulty. We know where the search for some sort of special essence can and does lead, an essence that would at once not be an idea and not material reality, but would constitute their common substance, the ‘third’ that appears one time as an idea and another time as being. For an idea and being are mutually exclusive concepts. That which is an idea is not being, and vice versa. How, then, in general, can they be compared? In what, in general, can the basis of their interaction be, what is that in which they are ‘one and the same’?

This difficulty was sharply expressed in its naked logical form by Descartes. In its general form it is the central problem of any philosophy whatsoever, the problem of the relationship of ‘thought’ to the reality existing outside it and independently of it, to the world of things in space and time, the problem of the coincidence of the forms of thought and reality, i.e. the problem of truth or, to put it in traditional philosophical language, the ‘problem of the identity of thought and being’.

It is clear to everyone that ‘thought’ and ‘things outside thought’ are far from being one and the same. It is not necessary to be a philosopher to understand that. Everyone knows that it is one thing to have a hundred roubles (or pounds, or dollars) in one’s pocket, and another to have them only in one’s dreams, only in one’s thoughts. The concept obviously is only a state of the special substance that fills the brain box (we could go on, furthermore, explaining this substance as brain tissue or even as the very thin ether of the soul keeping house there, as the structure of the brain tissue, or even as the formal structure of inner speech, in the form of which thinking takes place inside the head); but the subject is outside the head, in the space beyond the head, and is something quite other than the internal state of thought, ideas, the brain, speech, etc.

In order to understand such self-evident things clearly, and to take them into consideration, it is not generally necessary to have Descartes’ mind; but it is necessary to have its analytical rigour in order to define the fact that thought and the world of things in space are not only and not simply different phenomena, but are also directly opposite.

Descartes’ clear, consistent intellect is especially needed in order to grasp the problem arising from this difficulty, namely, in what way do these two worlds (i.e. the world of concepts, of the inner states of thought, on the one hand, and the world of things in external space, on the other hand) nevertheless agree with one another?

Descartes expressed the difficulty as follows. If the existence of things is determined through their extension and if the spatial, geometric forms of things are the sole objective forms of their existence outside the subject, then thinking is not disclosed simply through its description in forms of space. The spatial characteristic of thinking in general has no relation to its specific nature. The nature of thinking is disclosed through concepts that have nothing in common with the expression of any kind of spatial, geometric image. He also expressed this view in the following way: thought and extension are really two different substances, and a substance is that which exists and is defined only through itself and not through something else. There is nothing common between thought and extension that could be expressed in a special definition. In other words, in a series of definitions of thought there is not a single attribute that could be part of the definition of extension, and vice versa. But if there is no such common attribute it is also impossible to deduce being rationally from thought, and vice versa, because deduction requires a ‘mean term’, i.e. a term such as might be included in the series of definitions of the idea and of the existence of things outside consciousness, outside thought. Thought and being cannot in general come into contact with one another, since their boundary (the line or even the point of contact) would then also be exactly that which simultaneously both divides them and unites them.

In view of the absence of such a boundary, thought cannot limit the extended thing, nor the thing the mental expression. They are free, as it were, to penetrate and permeate each other, nowhere encountering a boundary. Thought as such cannot interact with the extended thing, nor the thing with thought; each revolves within itself.

Immediately a problem arises: how then are thought and bodily functions united in the human individual? That they are linked is an obvious fact. Man can consciously control his spatially determined body among other such bodies, his mental impulses are transformed into spatial movements, and the movements of bodies, causing alterations in the human organism (sensations) are transformed into mental images. That means that thought and the extended body interact in some way after all. But how? What is the nature of the interaction? How do they determine, i.e. delimit, each other?

How does it come about that a trajectory, drawn by thought in the plane of the imagination, for example a curve described in its equation, proves to be congruent with the geometrical contours of the same curve in real space? It means that the form of the curve in thought (i.e. in the form of the ‘magnitude’ of the algebraic signs of the equation) is identical with a corresponding curve in real space, i.e. a curve drawn on paper in a space outside the head. It is surely one and the same curve, only the one is in thought and the other in real space; therefore, acting in accordance with thought (understood as the sense of words or signs), I simultaneously act in the strictest accord with the shape (in this case the geometrical contour) of a thing outside thought.

How can that be, if ‘the thing in thought’ and ‘the thing outside thought’ are not only ‘different’ but are also absolutely opposite? For absolutely opposite means exactly this: not having anything in ‘common’ between them, nothing identical, not one attribute that could at once be a criterion of the concept ‘thing outside thought’ and of the concept ‘thing in thought’, or ‘imagined thing’. How then can the two worlds conform with one another? And, moreover, not accidentally, but systematically and regularly, these two worlds that have absolutely nothing in common, nothing identical? That is the problem around which all Cartesians spin, Descartes himself, and Geulincx, and Malebranche, and the mass of their followers.

Malebranche expressed the principal difficulty arising here in his own witty way, as follows: during the siege of Vienna, the defenders of the city undoubtedly saw the Turkish army as ‘transcendental Turks’, but those killed were very real Turks. The difficulty here is clear; and from the Cartesian point of view on thought it is absolutely insoluble, because the defenders of Vienna acted, i.e. aimed and fired their cannonballs in accordance with the image of Turks that they had in their brains, in accordance with ‘imagined’, ‘transcendental Turks’, and with trajectories calculated in their brains; and the shots fell among real Turks in a space that was not only outside their skulls, but also outside the walls of the fortress.

How does it come about that two worlds having absolutely nothing in common between them are in agreement, namely the world ‘thought of’, the world in thought, and the real world, the world in space? And why? God knows, answered Descartes, and Malebranche, and Geulincx; from our point of view it is inexplicable. Only God can explain this fact. He makes the two opposing worlds agree. The concept ‘God’ comes in here as a ‘theoretical’ construction by which to express the obvious but quite inconceivable fact of the unity, congruence, and identity perhaps, of phenomena that are absolutely contrary by definition. God is the ‘third’ which, as the ‘connecting link’, unites and brings into agreement thought and being, ‘soul’ and ‘body’, ‘concept’ and ‘object’, action in the plane of signs and words and action in the plane of real, geometrically defined bodies outside the head.

Having come directly up against the naked dialectical fact that ‘thought’ and ‘being outside thought’ are in absolute opposition, yet are nevertheless in agreement with one another, in unity, in inseparable and necessary interconnection and interaction (and thus subordinated to some higher law – and moreover, one and the same law), the Cartesian school capitulated before theology and put the inexplicable (from their point of view) fact down to God, and explained it by a ‘miracle’, i.e. by the direct intervention of supernatural powers in the causal chain of natural events.

Descartes, the founder of analytical geometry, could therefore not explain in any rational way whatever the reason for the algebraic expression of a curve by means of an equation ‘corresponding’ to the spatial image of this curve in a drawing. They could not, indeed, manage without God, because according to Descartes, actions with signs and on the basis of signs, in accordance only with signs (with their mathematical sense), i.e. actions in the ether of ‘pure thought’, had nothing in common with real bodily actions in the sphere of spatially determined things, in accordance with their real contours. The first were pure actions of the soul (or thinking as such), the second – actions of the body repeating the contours (spatially geometric outlines) of external bodies, and therefore wholly governed by the laws of the ‘external’, spatially material world.

(This problem is posed no less sharply today by the ‘philosophy of mathematics’. If mathematical constructions are treated as constructions of the creative intellect of mathematicians, ‘free’ of any external determination and worked out exclusively by ‘logical’ rules – and the mathematicians themselves, following Descartes, are quite often apt to interpret them precisely so – it becomes quite enigmatic and inexplicable why on earth the empirical facts, the facts of ‘external experience’, keep on agreeing and coinciding in their mathematical, numerical expressions with the results obtained by purely logical calculations and by the ‘pure’ actions of the intellect. It is absolutely unclear. Only ‘God’ can help.)

In other words the identity of these absolute opposites (‘thought’, ‘spirit’, and ‘extension’, ‘body’) was also recognised by Descartes as a factual principle – without it even his idea of an analytical geometry would have been impossible (and not only inexplicable) – but it was explained by an act of God, by his intervention in the interrelations of ‘thought and being’, ‘soul and body’. God, moreover, in Cartesian philosophy, and especially for Malebranche and Geulincx, could be understood as the purely traditional Catholic, orthodox God, ruling both the ‘bodies’ and the ‘souls’ of men from outside, from the heights of his heavenly throne, and co-ordinating the actions of the ‘soul’ with those of the ‘body’.

Such is the essence of the famous psychophysical problem, in which it is not difficult to see the specifically concrete and therefore historically limited formulation of the central problem of philosophy. The problem of the theoretical understanding of thought (logic), consequently, and hence not of the rules of operating with words or other signs, comes down to solving the cardinal problems of philosophy, or of metaphysics, to put it in a rather old-fashioned way. And that assumes mastering the culture of the genuinely theoretical thinking represented by the classical philosophers, who not only knew how to pose problems with maximum clarity, but also knew how to solve them.